Distributive Property (Multiplying a monomial by a polynomial)

The distributive property is written as follows: a(b+c)=ab+ac

This property has many applications, but it is particularly valuable to help us multiply a monomial by a polynomial. For example, x(3x+5). Since there are variables involved, we cannot add what is in parenthesis first (remember, 3x and 5 are not like terms). Instead, we will use the distributive property to multiply.

The best way to use the distributive property is to remember these three steps:

1) Multiply the outside term by the first term in parenthesis 2) Put a plus sign 3) Multiply the outside term by the second term in parenthesis

Let's look at few examples

1) x(3x+5)=3x2+5x

Step 1: Multiply the outside term by the first term in parenthesis x.3x=3x2

Step 2: Put a plus sign

Step 3: Multiply the outside term by the second term in parenthesis: x.5=5x

The answer cannot be simplified because there are no like terms, and it is in standard form, so we are finished. Final answer: 3x2+5x

2) 2y(y-8)=2y2+(-16y)=2y2-16y

Step 1: Multiply the outside term by the first term in parenthesis 2y.y=2y2

Step 2: Put a plus sign

Step 3: Multiply the outside term by the second term in parenthesis: 2y(-8)=-16y

This could be our final answer, but the plus sign isn't needed in this problem, so we could rewrite it as 2y2-16y.

3) 3x2 (5x2-4x+2)=15x4+(-12x3 )+6x2=15x4-12x3+6x2

Step 1: Multiply the outside term by the first term in parenthesis 3x2.5x2=15x4

Step 2: Put a plus sign

Step 3: Multiply the outside term by the second term in parenthesis: 3x2 (-4x)=-12x3
This problem has a third term inside parenthesis, so we will just continue the pattern:

Step 4: Put a plus sign

Step 5: Multiply the outside term by the third term in parenthesis: 3x2 (2)=6x2

This could be our final answer, but the first plus sign isn't needed in this problem, so we could rewrite it as 15x4-12x3+6x2.